Let's analyze the statements regarding the graph of the function \( f(x) = \log_8(x) \).
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It has the x-intercept (1, 0):
- To find the x-intercept, we set \( f(x) = 0 \).
- \( \log_8(x) = 0 \) implies \( x = 8^0 = 1 \).
- Thus, this statement is true.
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It has the y-intercept (0, 1):
- The y-intercept occurs when \( x = 0 \), but the logarithm function \( \log_8(x) \) is not defined for \( x \leq 0 \).
- Therefore, this statement is false.
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It has a horizontal asymptote at y = 0 (the x-axis):
- As \( x \) approaches \( 0 \) from the right, \( \log_8(x) \) approaches \( -\infty \). It does not approach \( 0 \) at any end behavior; instead, it continues downward.
- Thus, this statement is false.
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It has a vertical asymptote at x = 0 (the y-axis):
- The graph of \( \log_8(x) \) approaches \( -\infty \) as \( x \) approaches \( 0 \) from the right, indicating a vertical asymptote at \( x = 0 \).
- Hence, this statement is true.
Summary of Statements
- True: It has the x-intercept (1, 0).
- False: It has the y-intercept (0, 1).
- False: It has a horizontal asymptote at y = 0 (the x-axis).
- True: It has a vertical asymptote at x = 0 (the y-axis).
Final Answers
The true statements are:
- It has the x-intercept (1, 0).
- It has a vertical asymptote at x = 0 (the y-axis).